Austria 2023 Final Round Problem 1

Let $\alpha$ be a nonzero real number. Determine all functions $f : \mathbb{R} → \mathbb{R}$ with $$f(f(x + y)) = f(x + y) + f(x)f(y) + \alpha xy$$ for all $x, y \in \mathbb{R}$.

import Mathlib.Data.Real.Basic

def P (f : ℝ → ℝ) (α : ℝ) := ∀ x y, f (f (x + y)) = f (x + y) + f x * f y + α * x * y

def answer (α : ℝ) : Set (ℝ → ℝ) := sorry

theorem solution (f : ℝ → ℝ) {α : ℝ} (hα : α ≠ 0) : P f α ↔ f ∈ answer α := sorry

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